GAP TAILS AND FRAGMENTATION IN ONE DIMENSION: An exact reduction for Poisson and GUE proximity graphs

Abstract

Exact reduction of one-dimensional hard-radius proximity graphs to consecutive-gap exceedances. The result gives closed Poisson benchmarks and identifies a GUE–Poisson fragmentation crossover near one mean spacing. Numerical GUE experiments confirm the two-regime structure and separate local fragmentation from whole-graph connectivity.